NationMaster - Encyclopedia: Coordinate vector

In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fⁿ. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers (matrices and column vectors), which we know how to do explicitly. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-dimensional vector space over a field F. Definition Let F denote an arbitrary field (such as the real numbers R or the complex numbers C). ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In linear algebra, a column vector is an m Ã— 1 matrix, i. ...

1 Definition
2 The standard representation
3 Examples
- 3.1 Example 1
- 3.2 Example 2
4 Basis transformation matrix

Definition

Let V be a vector space of dimension n over a field F and let In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

$B = { b_1, b_2, ldots, b_n }$

be an ordered basis for V. Then for every $v in V$ there is a unique linear combination of the basis vectors that equals v: In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

$v = alpha _1 b_1 + alpha _2 b_2 + cdots + alpha _n b_n$

By one of the defining properties of bases, the α-s are determined uniquely by v and B. Now, we define the coordinate vector of v relative to B (also called the B representation of v) to be the following column vector:

$[ v ]_B = begin{bmatrix} alpha _1 vdots alpha _n end{bmatrix}.$

The α-s are called the coordinates of v. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

The standard representation

We can mechanize the above transformation by defining a function $φ B$ , called the standard representation of V with respect to B, that takes every vector to its coordinate representation: $φ B (v) = [v] B$ . Then $φ B$ is a linear transformation from V to Fⁿ. In fact, it is an isomorphism, and its inverse $phi_B^{-1}:mathbf{F}^nto V$ is simply In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

$phi_B^{-1}(alpha_1,ldots,alpha_n)=alpha_1 b_1+cdots+alpha_n b_n.$

Alternatively, we could have defined $phi_B^{-1}$ to be the above function from the beginning, realized that $phi_B^{-1}$ is an isomorphism, and defined $φ B$ to be its inverse.

Examples

Example 1

Let P3 be the space of all the algebraic polynomials in degree less than 4 (i.e. the highest exponent of x can be 3). This space is linear and spanned by the following polynomials: In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

B P = {1, x, x 2, x 3}

matching

$1 := begin{bmatrix} 1 0 0 0 end{bmatrix} quad ; quad x := begin{bmatrix} 0 1 0 0 end{bmatrix} quad ; quad x^2 := begin{bmatrix} 0 0 1 0 end{bmatrix} quad ; quad x^3 := begin{bmatrix} 0 0 0 1 end{bmatrix} quad$

then the corresponding coordinate vector to the polynomial

$p left( x right) = a_0 + a_1 x + a_2 x^2 + a_3 x^3$ is $begin{bmatrix} a_0 a_1 a_2 a_3 end{bmatrix}$ .

According to that representation, the differentiation operator d/dx which we shall mark D will be represented by the following matrix: Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... For the square matrix section, see square matrix. ...

$Dp(x) = P'(x) quad ; quad [D] = begin{bmatrix} 0 & 1 & 0 & 0 0 & 0 & 2 & 0 0 & 0 & 0 & 3 0 & 0 & 0 & 0 end{bmatrix}$

Using that method it is easy to explore the properties of the operator: such as invertibility, hermitian or anti-hermitian or none, spectrum and eigenvalues and more. In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

Example 2

The Pauli matrices which represent the spin operator when transforming the spin eigenstates into vector coordinates. The Pauli matrices are a set of 2 Ã— 2 complex Hermitian and unitary matrices. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

Basis transformation matrix

Let's mark with [M]^B the matrix which has columns consisting of b₁, b₂, ..., b_n . Then, In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

v = [M] B [v] B

This formalism can be generalized for transforming v from B representation to a C representation (where C is another basis). Defining basis transformation matrix from B to C as the following matrix:

$[M]_{C}^{B} = begin{bmatrix} [b_1]_C & cdots & [b_n]_C end{bmatrix}$

we receive the following theorem: Look up theorem in Wiktionary, the free dictionary. ...

$[v]_C = [M]_{C}^{B} [v]_B$

Corollary:
This matrix is Invertible matrix and M^-1 is the basis transformation matrix from C to B. In other words, In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

$[M]_{C}^{B} [M]_{B}^{C} = [M]_{C}^{C} = mathrm{Id}$

$[M]_{B}^{C} [M]_{C}^{B} = [M]_{B}^{B} = mathrm{Id}$

Remarks:

The basis transformation matrix can be regarded as an automorphism over V.
where E is the standard basis.
In order to easily remember the theorem

$[v]_C = [M]_{C}^{B} [v]_B$

notice that the M's sub-index and v's sub-index are "canceling" each other and the M's sub-index is what remains and become v's new sub-index. The "canceling" of index is not a real canceling but rather a manipulation of symbols which serves us for purposes of convenience.

Category: Linear algebra In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ... Look up theorem in Wiktionary, the free dictionary. ...

Results from FactBites:

Coordinate vector (527 words)
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